An Exact Logarithmic-State Algorithm for Recursive Tight-Cycle Constructions

The classical recursive construction for tight-cycle-free three-uniform hypergraphs has optimal edge count R(n) = max_{0 <= m < n} {m binom(n-m,2) + R(m)}. We give an exact algorithm for a single input n that visits only O(log(n+2)) states. A square-completed cubic potential shows that every maximizing split is one of the two integers surrounding ((2n+1)sqrt(3)-(2n+3))/4. Their descendants occupy at most three consecutive integers per depth and halve in size. The same identity gives an explicit linear error bound and R(n) = (sqrt(3)/3-1/2)n^3 + (sqrt(3)-2)n^2/2 + O(n). Integer square roots suffice for exact evaluation and reconstruction of an optimal block sequence. Nearest-integer rounding alone fails at n=106. The construction, recurrence, leading density and finite directed-star interpretation are classical and credited. The single-forbidden-tight-five-cycle Turan density conjecture is not resolved. This is an unrefereed preprint; no absolute priority is claimed.